We record and substantially extend what is known about the closed forms for various Bessel function moments arising in quantum field theory, condensed matter theory and other parts of mathematical physics. In particular, we develop formulae for integrals of products of six or fewer Bessel functions. In consequence, we are able to discover and prove closed forms for c_n,k := ∫^∞₀t^kKⁿ₀(t) dt with integers n = 1, 2, 3, 4 and k ≥ 0, obtaining new results for the even moments c_3,2k and c_4,2k. We also derive new closed forms for the odd moments s_n,2k+1 := ∫^∞₀t^2k+1I₀(t)Kⁿ⁻¹₀(t) dt with n = 3, 4 and for t_n,2k+1 := ∫^∞₀t^2k+1I²₀(t)Kⁿ⁻²₀(t) dt with n = 5, relating the latter to Green functions on hexagonal, diamond and cubic lattices. We conjecture the values of s_5,2k+1, make substantial progress on the evaluation of c_5,2k+1, s_6,2k+1 and t_6,2k+1 and report more limited progress regarding c_5,2k, c_6,2k+1 and c_6,2k. In the process, we obtain eight conjectural evaluations, each of which has been checked to 1200 decimal places. One of these lies deep in four-dimensional quantum field theory and two are probably provable by delicate combinatorics. There remains a hard core of five conjectures whose proofs would be most instructive, to mathematicians and physicists alike.